Camera Path Constraint not working as wanted
What you say there:


  nothing moves except a dashed blue line follows the path in the exact
  way I'd want the camera to


Makes me believe that you have the Influence set to 0.

Here a visual version of what i could imagine you have.


The camera or whatever object get's the follow path constraint, follows with it's position relative to the start-point on the path. Since it's not key-framed, it's still possible to adjust the location to suit your needs.

A suggestion i would want to give you is that it might be a better idea to parent the camera to an empty and let the empty follow the path instead.
That way you can give the camera it's own constraint, for example to target an empty for easier targeting. If you then parent the target-empty to an object, it's possible to slowly focus onto the object stay with it while in motion and eventually even wander around on the object, to then finally move away from it to close a scene. (hope that tip did not burst the topic)

Happy Blending.